Snap vertex without deforming mesh
I agree it would be nice to be able to enter a distance value after pressing G twice.
Method 1
Select the edge your vertex is on, and press Shift Numpad 1, 3 or 7, depending on the situation.
Then set the Transform Orientation to View, press the key to move along the desired axis (X,Y or Z) and enter your distance value.
Method 2
Select the vertex you want to move. press Shift + S and choose Cursor to Selected. Then press Shift + D to duplicate, but don't move it. Then extrude it in a a direction to make a helper edge (along a main axis, close to the edge you want the vertex to move along is probably best) and while doing this, type the exact distance you would want to move your vertex.
Now go to the snapping menu and choose Snap to Edge, check the box Align Rotation to Target, and for the Affect choose Rotate.
Then as Transform Pivot Point choose 3D Cursor.
Now rotate your helper edge, and it will snap to (and align with) the edge your vertex is on. The end of the helper edge will be at right distance. Now you can snap your vertex to that point and delete the helper edge afterwards.
Be sure though, before snapping the vertex, to go back to the snapping menu and choose Snap to Vertex, and for the Affect choose Move. (unchecking the box Align Rotation to Target is not needed but does affect the way snapping responds)